Finding out the coordinates of the right angle in a right triangle, knowing only the other two coordinates and angle relative to the plane

Question

I'm faced with the situation as presented on this picture.

I know the coordinates of $A$ and $B$. And I know that the vectors $AC$ and $CB$ lay in a $45^{\circ}$ angle relative to the plane (as shown on the picture).

I know that if not for the $45^{\circ}$ condition, there would be infinite solutions for $C$. But with that condition, there should be only two solutions. How do I find them?

Answer 1

You know the coordinates of point $A$ and $B$. Let $A=(x_a,y_a)$ and $B=(x_b,y_b)$. Now slope of $AC$ is $-1$ and that of $BC$ is 1. Then the equations of lines corresponding to the segments $AC$ and $BC$ may be obtained:
$y-y_a=-1(x-x_a)$
$y-y_b=1(x-x_b)$
Then, $C$ is then the intersection of the above lines.